This open access book explores commentaries on an influential text of pre-Copernican astronomy in Europe. It features essays that take a close look at key intellectuals and how they engaged with the main ideas of this qualitative introduction to geocentric cosmology. Johannes de Sacrobosco compiled his Tractatus de sphaera during the thirteenth century in the frame of his teaching activities at the then recently founded University of Paris. It soon became a mandatory text all over Europe. As a result, a tradition of commentaries to the text was soon established and flourished until the second half of the 17th century. Here, readers will find an informative overview of these commentaries complete with a rich context. The essays explore the educational and social backgrounds of the writers. They also detail how their careers developed after the publication of their commentaries, the institutions and patrons they were affiliated with, what their agenda was, and whether and how they actually accomplished it. The editor of this collection considers these scientific commentaries as genuine scientific works. The contributors investigate them here not only in reference to the work on which it comments but also, and especially, as independent scientific contributions that are socially, institutionally, and intellectually contextualized around their authors.

Originally published in 1926, this textbook was aimed at first-year undergraduates studying physics and chemistry, to help them become acquainted with the concepts and processes of differentiation and integration. Notably, a prominence is given to inequalities and more specifically to inequations, as reflected in the syllabus and general practice of the time. The book is divided into four parts: 'Number', 'Logarithms', 'Functions' and 'Differential and integral calculus'. Appendices are included as well as biographical notes on the mathematicians mentioned and an index of symbols. A self-contained and systematic introduction on mathematical analysis, this book provides an excellent overview of the essential mathematical theorems and will be of great value to scholars of the history of education.

This is a new annotated edition of Thomas J. Stieltjes' Collected Papers, first published in 1914 (Vol. I) and 1918 (Vol. II) by Noordhoff, Groningen, in French, and now published by Springer-Verlag, originally to mark the occasion of the 100th anniversary of Stieltjes' death (1894). These two volumes will be of great interest to all mathematicians who are anxious to understand the impact of Stieltjes' work on modern mathematics, and in particular on the theory of orthogonal polynomials and continued fractions. In addition to the reproduction of Stieltjes' papers (I-XLVII), Volume I includes about 75 pages of commentaries by contemporary mathematicians on Stieltjes' work. Volume II contains Stieltjes' papers XLVIII-LXXXIV together with English translations of his main paper "Recherches sur les fractions continues" and his short note regarding the Riemann hypothesis. A Bibliography of Stieltjes' papers is included in both volumes for the convenience of the reader.

Joseph Larmour (1857-1942) was a theoretical physicist who made important discoveries in relation to the electron theory of matter, as espoused in his 1900 work Aether and Matter. Originally published in 1929, this is the first part of a two-volume set containing Larmour's collected papers. The papers are presented in chronological order across the volumes, enabling readers to understand their theoretical development and framing them in an accessible form for 'future historical interests'. Authorial notes and appendices are also included. This book will be of value to anyone with an interest in the word of Larmour, mathematics physics and the history of science.

Originally published in 1921, this book was intended as a textbook of dynamics for the use of students who have some acquaintance with the methods of the differential and integral calculus. The chapters cover a vast range of topics and include the existing well-known key theorems of the day; chapters include, 'Displacement, velocity, acceleration', 'Forces acting on a particle' and 'The rotation of the Earth'. Notably, difficult and challenging topics are marked with an asterisk to indicate the advanced nature of the subject and a collection of miscellaneous examples are appended to most of the chapters to assist with classes and revision, most of which have been sourced from previous examination papers. Linear equations and diagrams are included throughout to support the text. This book will be a valuable resource to scholars of physics and engineering as well as to anyone with an interest in the history of education.

First published in 1927, as the second edition of a 1915 original, this book presents exercises in arithmetic aimed at school students. The text is divided into three main sections: Part I mainly covers integers; Part II covers fractions; Part III covers miscellaneous areas. Each section ends with revision papers and more exercises. This book will be of value to anyone with an interest in mathematics and the history of education.

Joseph Larmour (1857-1942) was a theoretical physicist who made important discoveries in relation to the electron theory of matter, as espoused in his 1900 work Aether and Matter. Originally published in 1929, this is the second part of a two-volume set containing Larmour's collected papers. The papers are presented in chronological order across the volumes, enabling readers to understand their theoretical development and framing them in an accessible form for 'future historical interests'. Authorial notes and appendices are also included. This book will be of value to anyone with an interest in the word of Larmour, mathematics physics and the history of science.

Abraham Adrian Albert (1905-72) was an American mathematician primarily known for his groundbreaking work on algebra. In this book, which was originally published in 1938, Albert provides a detailed exposition of 'modern abstract algebra', taking into account numerous discoveries in the field during the preceding ten years. A glossary is included. This is a highly informative book that will be of value to anyone with an interest in the development of algebra and the history of mathematics.

In 1988 The Mathematical Intelligencer, a quarterly mathematics journal, carried out a poll to find the most beautiful theorem in mathematics. Twenty-four theorems were listed and readers were invited to award each a 'score for beauty'. While there were many worthy competitors, the winner was 'Euler's equation'. In 2004 Physics World carried out a similar poll of 'greatest equations', and found that among physicists Euler's mathematical result came second only to Maxwell's equations. The Stanford mathematician Keith Devlin reflected the feelings of many in describing it as "like a Shakespearian sonnet that captures the very essence of love, or a painting which brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence". What is it that makes Euler's identity, ei + 1 = 0, so special? In Euler's Pioneering Equation Robin Wilson shows how this simple, elegant, and profound formula links together perhaps the five most important numbers in mathematics, each associated with a story in themselves: the number 1, the basis of our counting system; the concept of zero, which was a major development in mathematics, and opened up the idea of negative numbers; an irrational number, the basis for the measurement of circles; the exponential e, associated with exponential growth and logarithms; and the imaginary number i, the square root of -1, the basis of complex numbers. Following a chapter on each of the elements, Robin Wilson discusses how the startling relationship between them was established, including the several near misses to the discovery of the formula.

This revealing work examines an approach from ancient astronomy to what was then a particularly important question, namely that of understanding the relationship between the position in the ecliptic and the time it takes for a fixed-length of the ecliptic beginning at that point to rise above the eastern horizon. Schemes known as "rising time schemes" were used to give lengths of the celestial equator corresponding to each of the twelve zodiacal signs which make up the ecliptic. This book investigates the earliest known examples of these schemes which come from Babylonia and date to the mid to late first millennium BC. Making an important contribution to our knowledge of astronomy in the ancient world, this volume includes editions and translations of all of the known Babylonian rising time texts, including several texts that are identified for the first time. Through a close examination of the preserved texts it has been possible to reconstruct the complete Babylonian rising time scheme. This reconstruction is unprecedented in its completeness, and it is also now possible to situate the scheme within a genre of Babylonian astronomy known as schematic astronomy which presents theoretical descriptions of the astronomical phenomena. The unique discoveries and fresh explorations in this book will be of interest to historians of ancient astronomy, scholars of Babylonian history and those investigating the origins of scientific thought.

An important figure in the development of modern mathematical logic and abstract algebra, Augustus De Morgan (1806-71) was also a witty writer who made a hobby of collecting evidence of paradoxical and illogical thinking from historical sources as well as contemporary pamphlets and periodicals. Based on articles that had appeared in The Athenaeum during his lifetime, this work was edited by his widow and published in book form in 1872. It parades all varieties of crackpot, from circle-squarers to inventors of perpetual motion machines, all for the reader's entertainment and education. Filled with anecdotes, personal opinions and 'squibs' of every kind, the book remains enjoyable reading for those who are amused rather than appalled by the human condition. Also reissued in the Cambridge Library Collection are the Memoir of Augustus De Morgan (1882), prepared by his wife, and his ambitious Formal Logic (1847).

Luis Antonio Santalo (Spain 1911 - Argentina 2001) contributed to several branches of Geometry, his laying of the mathematical foundations of Stereology and its applications perhaps being his most outstanding achievement. A considerable power of abstraction, a brilliant geometric intuition and an outstanding gift as a disseminator of science were among his virtues. The present volume contains a selection of his best papers. Part I consists of a short biography and some photographs together with a complete list of his publications, classified into research papers, books, and articles on education and the popularization of mathematics, as well as a comprehensive analysis of his contribution to science. Part II, the main part of the book, includes selected papers, arranged into five sections according to the nature of their contents: Differential Geometry, Integral Geometry, Convex Geometry, Affine Geometry, and Statistics and Stereology. Each section is preceded by a commentary by a renowned specialist: Teufel, Langevin, Schneider, Leichtweiss, and Cruz-Orive, respectively. Finally, Part III emphasizes the influence of his work. It contains commentaries by several specialists regarding modern results based on, or closely related to, those of Santalo, some book reviews written by Santalo, as well as some reviews of his books. As a curious addendum, a ranking of his articles, given by Santalo himself, is included.

Originally published in 1946, this book explains important aspects of the world through the lens of mathematics. McKay discusses important questions such as time, the size of the earth and 'numbers that mean too much' in language that is enthusiastic and easily accessible to non-mathematicians. This book will be of value to anyone with an interest in the history of mathematics.

Newton's Principia paints a picture of the earth as a spinning, gravitating ball. However, the earth is not completely rigid and the interplay of forces will modify its shape in subtle ways. Newton predicted a flattening at the poles, yet others disagreed. Plenty of books have described the expeditions which sought to measure the shape of the earth, but very little has appeared on the mathematics of a problem which remains of enduring interest even in an age of satellites. Published in 1874, this two-volume work by Isaac Todhunter (1820-84), perhaps the greatest Victorian historian of mathematics, takes the mathematical story from Newton, through the expeditions which settled the matter in Newton's favour, to the investigations of Laplace which opened a new era in mathematical physics. Volume 1 traces developments from Newton up to 1780, including coverage of the work of Maupertuis, Clairaut and d'Alembert.

Newton's Principia paints a picture of the earth as a spinning, gravitating ball. However, the earth is not completely rigid and the interplay of forces will modify its shape in subtle ways. Newton predicted a flattening at the poles, yet others disagreed. Plenty of books have described the expeditions which sought to measure the shape of the earth, but very little has appeared on the mathematics of a problem, which remains of enduring interest even in an age of satellites. Published in 1874, this two-volume work by Isaac Todhunter (1820-84), perhaps the greatest Victorian historian of mathematics, takes the mathematical story from Newton, through the expeditions which settled the matter in Newton's favour, to the investigations of Laplace which opened a new era in mathematical physics. Volume 2 is largely devoted to the work of Laplace, tracing developments up to 1825.

Originally published in 1911 as number thirteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book presents a general survey of the problem of the 27 lines upon the cubic surface. Illustrative figures and a bibliography are also included. This book will be of value to anyone with an interest in cubic surfaces and the history of mathematics.

First published in 1930, as the third edition of a 1907 original, this book forms number six in the Cambridge Tracts in Mathematics and Mathematical Physics Series. The text gives a concise account of the theory of equations according to the ideas of Galois. This book will be of value to anyone with an interest in algebra and the history of mathematics.

Originally published in 1910 as number eleven in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book deals with differential calculus and its underlying structures. Appendices on further reading and clarification of certain points are also included. This tract will be of value to anyone with an interest in the history of mathematics or calculus.

Originally published in 1910 as number twelve in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides an up-to-date version of Du Bois-Reymond's Infinitarcalcul by the celebrated English mathematician G. H. Hardy. This tract will be of value to anyone with an interest in the history of mathematics or the theory of functions.

Originally published in 1946 as number thirty-nine in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding linear groups. Appendices are also included. This book will be of value to anyone with an interest in linear groups and the history of mathematics.

Originally published in 1913 as number fourteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding the properties of the twisted cubic. A bibliography and appendix section are also included. This book will be of value to anyone with an interest in the twisted cubic and the history of mathematics.

Originally published in 1932 as number twenty=seven in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account of the theory of modular invariants as embodied in the work of Dickson, Glenn and Hazlett. Appendices are included. This book will be of value to anyone with an interest in modular invariants and the history of mathematics.

First published in 1913, as the second edition of a 1905 original, this book is the first volume in the Cambridge Tracts in Mathematics and Mathematical Physics Series. The text provides a concise account regarding volume and surface integrals used in physics. This book will be of value to anyone with an interest in integrals and physics.

Originally published in 1915 as number eighteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, and here reissued in its 1952 reprinted form, this book contains a condensed account of Dirichlet's Series, which relates to number theory. This tract will be of value to anyone with an interest in the history of mathematics or in the work of G. H. Hardy.

Originally published in 1908 as number nine in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding the invariant theory connected with a single quadratic differential form. This book will be of value to anyone with an interest in quadratic differential forms and the history of mathematics.